The Pentagonal Number Theorem

Again a little bit of a flashback to my earliest steps in doing mathematical research. The Pentagonal number theorem is one of the most beautiful theorems in number theory. It uses the Pentagonal numbers $f(n)=(3n^2-n)/2$ to get a recursion for the partition function $p(n) = \sum_{k \neq n} (-1)^k p(n-f(k))$ . This can be written as ( typo on the board) $\sum_{k \in \mathbb{Z}} (-1)^k p(n-f(k)) = 0$ . The usual way to write this is $p(n) = p(n-1)+p(n-2)-p(n-5)-p(n-7) + \dots$ which has a ++– pattern but using the pentagonal numbers on the integers makes the formula more elegant and sees it as a super sum. Euler derived this by looking at $\prod_{j=1}^{\infty} (1-z^j) = \sum_{k \in \mathbb{Z}} (-1)^k z^{f(k)}$ and its reciprocal $\prod_{j=1}^{\infty} (1-z^j)^{-1} p(k) z^k$ . In the first case, we see the Pentagonal numbers, in the second we see the Partition numbers.

Around 2008, while working with John Lesieutre about almost periodic Dirichlet series the topic of Birkhoff sums $\sum_{k=1}^{\infty} g(k \alpha)$ came up. This can be studied in a bit more general frame work by looking at almost periodic Taylor series $\sum_{k=1}^{\infty} g(k \alpha) z^k$ of almost periodic Dirichlet series $\sum_{k=1}^\infty g(k \alpha)/k^s$ . We proved that if $\alpha$ is Diophantine and $g$ is smooth, then the almost periodic Dirichlet series has an analytic continuation to the entire complex plane. What happens if g is no more smooth? With Folkert Tangerman, we looked at the case $g(x) =\log([2-2 \cos(2 \pi x) = 2 \log|1-e^{2\pi i x}|$ . See the paper “Self-similarity and growth in Birkhoff sums“. Writing $z=e^{2\pi i \alpha}$ . If $S(n)=\sum_{k=1}^n g(k \alpha)$ , then $e^{S(n)/2} = \prod_{k=1}^n (1-z^k)$ and $e^{-S(n)/2} = \prod_{k=1}^n (1-z^k)^{-1}$ . These are finite product versions of the infinite product. For more, see the golden rotations summary. Especially interesting is the Birkhoff sum of the cotangent function. It is a random walk with almost periodically correlated random variables which have infinite variance. What happens there is that the random walk has a fractal almost periodic limit. See the paper “Selfsimilarity in the Birkhoff sum of the cotangent function”. The topic illustrates that sometimes, it can be interesting to look at very special problems like picking the most natural real number “the golden ratio” (constant continued fraction expansion 11111) and the most natural periodic function (constant Fourier series 1111111 “the cotangent function” and combine them. Magic happens because the random walk has now a self-similar structure: for IID random variables $X_k$ , the random walk is boring: by the law of large numbers $f_n(x) = S_[x n]/n$ converges for $n \to \infty$ to the linear function $f(x)=x E[X]$ for $x \in [0,1]$ and almost all $\omega \in \Omega$ , the underlying probability space. With correlations but with ergodicity and finite variance, this still is boring: Birkhoffs ergodic theorem still assures that we converge to that boring f(x)=x growth, almost everywhere. The magic starts when we do not have finite variance, nor have de-correlated random variables any more. And the cotangent function is the prototype where $X_k = \cot(k \alpha)$ produces random variables which have a Cauchy distribution. There is still a central limit theorem (I talked about this on my youtube channel last year for a couple of Saturdays in the context of high risk Stochastic processes but realized then that all this has been known since Levy and Kolmogorov!) In the almost periodic, infinite variance case (a topic I got into with work with Tangermann), the limiting function $\lim_{n \to \infty} S_[x n]/n$ can become much more interesting. In the case when the random variables are $X_k(\omega)= \cot(\omega + k \alpha)$ (with starting point $\omega=0$ with golden ration $\alpha$ , there is an explicit fractal limiting function. That’s what I called the “Golden Graph”.

Author: oliverknill